Density enhancement mechanism of upwind schemes for low Mach number flows

Lin, Boxi; Yan, Chao; Chen, Shusheng

doi:10.1007/s10409-017-0737-9

Cited by 8 publications

(12 citation statements)

References 39 publications

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“…The discrete single-timescale expansion with the acoustic diffusion can be found in a several papers, as it shows why classical transonic schemes fail at low Mach number, see [25][26][27]31]. Several of these papers also present the discrete single-timescale expansion with the mixed diffusion, as this is the most commonly reached diffusion scaling for low Mach number fixes to transonic schemes [27,30,31,34,37,44]. To the author's knowledge, the only previous study in the literature to carry out multiple-timescale expansions of the discrete equations is Bruel et al [33], who present multiple-timescale expansions of the baratropic Euler equations with all three schemes.…”

Section: Expansion Of the Discrete Euler Equations With Artificial Di...mentioning

confidence: 97%

“…The right-hand sides of (44a,44c) are in the form of the discrete Laplacian. In N dimensions, equations (44) are two elliptic systems each of N + 2 equations for p (0, 1) , which are the discrete equivalents of the continuous relations (26a,27a) and (26b,27b). Under a small number of reasonable assumptions, they enforce constant p (0,1) over the entire domain [25].…”

Section: Expansion With Acoustic Diffusionmentioning

confidence: 99%

“…Since then, many papers have been published presenting different methods for modifying the artificial diffusion to achieve accurate low Mach number results. These use a variety of strategies, building on central schemes [15,[21][22][23], flux-difference [7,20,[24][25][26][27][28][29][30][31][32][33][34][35][36][37][38], and flux-vector splitting [39][40][41][42][43][44] methods. The majority of studies consider only the convective limit, although a number also address flows with acoustic features [7, 21-23, 33, 42, 43, 45].…”

Section: Introductionmentioning

confidence: 99%

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Artificial diffusion for convective and acoustic low Mach number flows I: Analysis of the modified equations, and application to Roe-type schemes

Hope-Collins¹,

Mare²

2021

Preprint

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Three asymptotic limits exist for the Euler equations at low Mach number -purely convective, purely acoustic, and mixed convective-acoustic. Standard collocated density-based numerical schemes for compressible flow are known to fail at low Mach number due to the incorrect asymptotic scaling of the artificial diffusion. Previous studies of this class of schemes have shown a variety of behaviours across the different limits and proposed guidelines for the design of low-Mach schemes. However, these studies have primarily focused on specific discretisations and/or only the convective limit.In this paper, we review the low-Mach behaviour using the modified equations -the continuous Euler equations augmented with artificial diffusion terms -which are representative of a wide range of schemes in this class. By considering both convective and acoustic effects, we show that three diffusion scalings naturally arise. Single-and multiple-scale asymptotic analysis of these scalings shows that many of the important low-Mach features of this class of schemes can be reproduced in a straightforward manner in the continuous setting.As an example, we show that many existing low-Mach Roe-type finite-volume schemes match one of these three scalings. Our analysis corroborates previous analysis of these schemes, and we are able to refine previous guidelines on the design of low-Mach schemes by including both convective and acoustic effects. Discrete analysis and numerical examples demonstrate the behaviour of minimal Roe-type schemes with each of the three scalings for convective, acoustic, and mixed flows.

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Section: Expansion Of the Discrete Euler Equations With Artificial Di...mentioning

confidence: 97%

Section: Expansion With Acoustic Diffusionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Artificial diffusion for convective and acoustic low Mach number flows I: Analysis of the modified equations, and application to Roe-type schemes

Hope-Collins¹,

Mare²

2021

Preprint

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“…None of the three terms in (31) match those in (28). The upwinding and δU term provide no diffusion on the pressure field, which only has (erroneous) diffusion from the δp term scaled by M 2 .…”

Section: Entropy Variables Formmentioning

confidence: 98%

“…The Toro-Vasquez splitting was proposed by Toro & Vasquez 2012 [59] in the context of developing a general framework for the discretisation of convection-pressure flux-vector splittings. Over the last decade, the Zha-Bilgen and Toro-Vasquez splittings have gained some attention for low Mach number flow [45,4,22,56,31,7], and it has been shown that accurate schemes for this regime can be produced from these splittings. We cover more detail on the literature for low Mach number schemes using convection-pressure flux-vector splittings in section 4.…”

Section: Introductionmentioning

confidence: 99%

Artificial diffusion for convective and acoustic low Mach number flows II: Application to Liou-Steffen, Zha-Bilgen and Toro-Vasquez convection-pressure flux splittings

Hope-Collins¹,

Mare²

2023

Preprint

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Liou-Steffen splitting (AUSM) schemes are popular for low Mach number simulations, however, like many numerical schemes for compressible flow they require careful modification to accurately resolve convective features in this regime. Previous analyses of these schemes usually focus only on a single discrete scheme at the convective limit, only considering flow with acoustic effects empirically, if at all. In our recent paper Hope-Collins & di Mare, 2023 we derived constraints on the artificial diffusion scaling of low Mach number schemes for flows both with and without acoustic effects, and applied this analysis to Roe-type finite-volume schemes. In this paper we form approximate diffusion matrices for the Liou-Steffen splitting, as well as the closely related Zha-Bilgen and Toro-Vasquez splittings. We use the constraints found in Hope-Collins & di Mare, 2023 to derive and analyse the required scaling of each splitting at low Mach number. By transforming the diffusion matrices to the entropy variables we can identify erroneous diffusion terms compared to the ideal form used in Hope-Collins & di Mare, 2023. These terms vanish asymptotically for the Liou-Steffen splitting, but result in spurious entropy generation for the Zha-Bilgen and Toro-Vasquez splittings unless a particular form of the interface pressure is used. Numerical examples for acoustic and convective flow verify the results of the analysis, and show the importance of considering the resolution of the entropy field when assessing schemes of this type.

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A new diffusion-regulated flux splitting method for compressible flows

Kalita

Sarmah

2019

Computers & Fluids

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Density enhancement mechanism of upwind schemes for low Mach number flows

Cited by 8 publications

References 39 publications

Artificial diffusion for convective and acoustic low Mach number flows I: Analysis of the modified equations, and application to Roe-type schemes

Artificial diffusion for convective and acoustic low Mach number flows I: Analysis of the modified equations, and application to Roe-type schemes

Artificial diffusion for convective and acoustic low Mach number flows II: Application to Liou-Steffen, Zha-Bilgen and Toro-Vasquez convection-pressure flux splittings

A new diffusion-regulated flux splitting method for compressible flows

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